Fig 1.
Illustration for the United Kingdom of the proposed warping method to compute the time warping s(t) and ratio r(t) relating the daily case incidence f(t) and the daily death count g(t) through the warping equation r(t)f(t) = g(t + s(t)). (A): f(t) in cyan and g(t) in red. (B): comparison of r(t)f(t) with g(t + s(t)) showing how well the warping equation is satisfied. (C): the estimated r(t) in cyan and s(t) in red. Both are constrained by the method to be smooth to avoid overfitting. The color shaded rectangles represent the time intervals of the first five SARS-CoV-2 pandemic waves (as described in [1]).
Table 1.
Results, obtained in [3], for the mean of the time from symptom onset to death, segmented by symptom onset date using Lognornal distributions.
Table 2.
Comparison, for the United Kingdom, of the death-rate per capita obtained in [1] for the first five SARS-CoV-2 pandemic waves and the average of r(t) in the same periods estimated using the case and deaths time series.
Fig 2.
For France, we use as time series f(t): the daily value of the weekly hospitalizations divided by 7 and g(t): the restored number of daily deaths using EpiInvert. In (A) we plot f(t) and g(t). In (B) we plot the estimated r(t)f(t) and g(t + s(t)). In (C) we plot r(t) and s(t).
Fig 3.
For France, we show the evolution, during the first epidemic year of the mortality rate in hospital obtained in [4] and the average of r(t) in the same periods showed in Fig 2.
Fig 4.
Same experiment as in Fig 2 but using wr = 10 (instead of the default value wr = 1000) and ws = 1 (instead of the default value ws = 10).
Fig 5.
We use as time series f(t): the restored number of daily cases in France using EpiInvert and g(t): the smooth values of daily cases provided by the OWID. In (A) we plot f(t) and g(t). In (B) we plot the estimated r(t)f(t) and g(t + s(t)). In (C) we plot r(t) and s(t).
Fig 6.
We use as time series g(t) the restored number of deaths in France using EpiInvert and f(t) is defined as where s(t) and r(t) are a simulated ground truth. In (A) we plot f(t) and g(t). In (B) we plot the estimated r(t)f(t) and g(t + s(t)) using the recovered values for r(t) and s(t). In (C) we plot the recovered values for r(t) and s(t).
Fig 7.
Comparison of the values of r(t) and s(t) estimated by the model and their ground truth using the simulated data.
Fig 8.
We use as time series g(t) the raw number of reported deaths in France and f(t) is defined as where s(t) and r(t) are a simulated ground truth. In (A) we plot f(t) and g(t). In (B) we plot the estimated r(t)f(t) and g(t+ s(t)) using the recovered values for r(t) and s(t). In (C) we plot the recovered values for r(t) and s(t).
Fig 9.
Normalized error (10) using the simulated data.
Fig 10.
A sanity check: verifying the transitivity of the estimations of r(t) and s(t) when three time series are involved: f1(t), the France incidence curve, f2(t), the number of new hospital admissions and f3(t) the number of new deaths. In (A) we plot r1,3(t) and r2,3(t + s1,2(t))r1,2(t) and in (B) we plot s1,3(t) and s1,2(t) + s2,3(t + s1,2(t)).
Fig 11.
We use as time series f(t): the restored number of daily deaths in Italy using EpiInvert and g(t): the restored number of daily deaths in France using EpiInvert. In (A) we plot f(t) and g(t). In (B) we plot the estimated r(t)f(t) and g(t + s(t)). In (C) we plot r(t) and s(t).
Fig 12.
Results obtained using as f(t) the weekly flu incidence and g(t) = maxt(T(t)) − T(t) where T(t) is the weekly average of the temperature in France. The black vertical lines correspond to the peaks of the flu incidence.
Fig 13.
RMSE between the true r(t) and the one obtained by our method divided by the mean of the true r(t), using different choices for the regularization weights wr and ws.
Fig 14.
RMSE between the true s(t) and the one obtained by our method using different choices for the regularization weights wr and ws.